1. A three-link manipulator with rotational joints and prismatic joint is shown in figure. Calculate the velocity of the tip of the arm as a Jacobian. Give the answer in two froms - in terms of frame \( \{3\} \) and also in terms of frame \( \{0\} \)
DH parameters
\begin{tabular}{|c|c|c|c|c|}
\hline Link & \( a_{i} \) & \( \alpha_{i} \) & \( d_{i} \) & \( \theta_{i} \) \\
\hline 1 & 0 & 0 & \( d_{1} \) & \( \theta_{1}^{*} \) \\
\hline 2 & 0 & 90 & \( d_{2}^{*} \) & 180 \\
\hline 3 & 0 & 0 & \( d_{3}^{*} \) & 0 \\
\hline
\end{tabular}
( \( { }^{*} \) is actuator parameters)
\[
{ }_{1}^{0} T=\left[\begin{array}{cccc}
c_{1} & -s_{1} & 0 & 0 \\
s_{1} & c_{1} & 0 & 0 \\
0 & 0 & 1 & d_{1} \\
0 & 0 & 0 & 1
\end{array}\right],{ }_{2}^{1} T=\left[\begin{array}{cccc}
-1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & d_{2} \\
0 & 0 & 0 & 1
\end{array}\right],{ }_{3}^{2} T=\left[\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & d_{2} \\
0 & 0 & 0 & 1
\end{array}\right]
\]
